key term - Point-Slope Form

Definition

The point-slope form is a way to express the equation of a line given the slope of the line and a single point on the line. It allows you to easily write the equation of a line without having to find the y-intercept first.

5 Must Know Facts For Your Next Test

1. The point-slope form of a linear equation is written as $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point on the line and $m$ is the slope of the line.
2. The point-slope form is useful when you are given a single point on the line and the slope, as it allows you to easily find the equation of the line without having to find the y-intercept.
3. To graph a line using the point-slope form, you can plot the given point and then use the slope to determine the direction and steepness of the line.
4. The point-slope form can be rearranged into the standard slope-intercept form ($y = mx + b$) by solving for $y$ in the equation.
5. Understanding the point-slope form is essential for finding the equation of a line when given a point and the slope, which is a common task in graphing linear equations.

Review Questions

• Explain how the point-slope form of a linear equation is related to the concept of slope.
  • The point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, directly incorporates the concept of slope. The slope, $m$, represents the rate of change between the $x$ and $y$ variables and is a key component of the equation. The point $(x_1, y_1)$ serves as a reference point, and the slope $m$ is used to determine the relationship between $x$ and $y$ for any other point on the line.
• Describe how the point-slope form can be used to find the equation of a line when given a point and the slope.
  • When you are given a point $(x_1, y_1)$ and the slope $m$ of a line, you can use the point-slope form to easily find the equation of the line. Simply plug the known values into the equation $y - y_1 = m(x - x_1)$, and then solve for $y$ to obtain the standard slope-intercept form of the equation, $y = mx + b$. This process allows you to determine the equation of the line without having to first find the y-intercept, making it a useful tool for graphing linear equations.
• Analyze how the point-slope form can be used to graph a linear equation in the coordinate plane.
  • The point-slope form provides the necessary information to graph a linear equation directly on the coordinate plane. By knowing a point $(x_1, y_1)$ that lies on the line and the slope $m$, you can plot the given point and then use the slope to determine the direction and steepness of the line. This allows you to sketch the line accurately, as the slope tells you how the line will rise or fall as you move from the known point in the positive or negative $x$ direction. The point-slope form, therefore, is a powerful tool for visualizing the graph of a linear equation.

Related terms

Slope:

The slope of a line is a measure of its steepness, indicating the rate of change between the x and y variables.

Linear Equation:

A linear equation is an equation of a straight line that can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Coordinate Plane:

The coordinate plane is a two-dimensional grid used to graph linear equations, where the x-axis represents the horizontal dimension and the y-axis represents the vertical dimension.